Using Theorem REF , we can choose the hyperparameters in L2A such that the average risk is controlled.
Our bound shares the same form as the classical EWAF bound of \(\frac{1}{1 - \exp (-\lambda )}(\lambda \delta + \frac{1}{T} \ln m)\)  (see e.g. Theorem 2.4 of [1]}) and converges at the same rate with respect to \(T\) .
The difference is in the constants: our bound is scaled by \(-1/c(\lambda , \tilde{\epsilon }, z)\)  and the latter is scaled by \(\frac{1}{1 - \exp (-\lambda )}\) .
To understand the impact of this difference, we plot the two bounds for various abstention costs and meta-learning rates in Figure REF .
We set \(m = 10\) , \(T = 50\) , and, for the sake of comparison, \(n = \infty \) .
We assume the maximum drift is \(V = 2\delta \) .
We find that the curvatures of our risk bounds are significantly flatter and allow for much larger learning rates.
For example, for abstention cost \(\delta = 0.15\) , the minimum attainable risk bound from the classical result is \(2\delta \)  and corresponds to using a learning rate of \(\lambda = 0.70\) .
We can achieve this same risk bound using Theorem REF  with a learning rate that is nearly three times as large, which translates to a much faster approval rate for beneficial modifications.
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